NEW CLASS OF SOFT LINEAR ALGEBRAIC CODES ANDTHEIR PROPERTIES USING SOFT SETS

MUMTAZ ALI, FLORENTIN SMARANDACHE, AND W.B. VASANTHA KANDASAMY

Abstract. Algebraic codes play a signi�cant role in the minimization of datacorruption which caused by defects such as inference, noise channel, crosstalk,and packet lost. In this paper, we introduced soft codes (soft linear codes)through the application of soft sets which is an approximated collection ofcodes. We also discussed several types of soft codes such as type-1 soft codes,complete soft codes etc. The innovative idea of soft codes is advantageous,for it can simultaneously transmit n-distinct messages to n-set of receivers.Further this new technique makes use of bi-matrices or to be more generaluses the concept of n-matrices. Certainly this notion will save both time andeconomy. Moreover, we develop two techniques for the decoding of soft codes.At the end, we present a soft communication process and develop a model forthis soft communication process. The disticntions and comparison of softlinear codes and linear codes are also presented.

1. Introduction

The transmission and storage of large amounts of data reliably and without erroris a signi�cant part of the modern communication systems. Algebraic codes areused for data compression, cryptography, error correction and for network coding.The theory of codes was �rst focused by Shanon in 1948 and then gradually de-veloped by time to time by di¤erent researchers. There are many types of codeswhich is important to its algebraic structures such as Linear block codes, Ham-ming codes, BCH codes [46] and so on. The most common type of code is a linearcode over the �eld Fq. Recently a variety of codes over �nite rings have been stud-ied. The linear codes over �nite rings are initiated by Blake in a series of papers[13; 14] and Spiegel [49; 50] : Huber de�ned codes over Gaussian integers [26; 27; 28].Shankar studied BCH codes over rings of residue integers [46]. Satyanarayana con-sider analyses of codes over Zn by viewing their properties under the Lee metric [44].Some more literature can be studied in [18; 19; 20; ; 21; 23; 30; 31; 35; 38; 43; 44; 46] :Zadeh in his seminal paper introduced the innovative concept of fuzzy sets in

1965 [53] . A fuzzy set is characterized by a membership function whose valuesare de�ned in the unit interval [0; 1] and thus fuzzy set perhaps is the most suit-able framework to model uncertain data. Fuzzy sets have a several interestingapplications in the areas such as signal processing, decision making, control theory,reasoning, pattern recognition, computer version and so on. The theory of fuzzyset is a signi�cantly used in medical diagnosis, social science, engineering etc. Thealgebraic structures in the context of fuzzy sets have been studied such as fuzzygroups, fuzzy rings, fuzzy semigroups fuzzy codes etc. Alcantud et al. studied

Key words and phrases. Code, linear code, generator matrix, parity check matrix, soft set, softcode, generator bimatrix, generator n-matrix, parity check bimatrix, parity check n-matrix. .

1

2 MUMTAZ ALI, FLORENTIN SMARANDACHE, AND W.B. VASANTHA KANDASAMY

real applications of hesitant fuzzy sets to improvement of teaching performanceassessments and construction of meta-rankings for ranking universities [4]. Gon-zalez et al. discussed application of interval type-2 fuzzy systems to edgedetection [25].Some more study on fuzzy set can be found in [52; 54].The theory of rough sets was �rst introduced by Pawlak in 1982 which is another

signi�cant mathematical tool to handle vague data and information [22; 39; 40] .The theory of rough sets mainly based upon equivalence classes to approximatecrisp sets. Rough sets has several applications in data mining, machine learning,medicine, data analysis, expert systems and cognitive analysis etc. Some moreliterature can be found on rough sets in [15; 17; 22; 29; 41; 42; 47; 51]. Algebraicstructures can also be studied in the context of rough set such as rough groups[12], rough semigroups and so on. Chen et al. gave the application of roughsets to graph theory [16].The complexities of modelling uncertain data is the main problem in engineer-

ing, environmental science, economics, social sciences, health and medical sciencesetc. Classical theories are not always successful as the uncertainties are of severaltypes which appearing in these domains. The fuzzy set theory [53], probabilitytheory, rough set theory [22; 39; 40] etc are well known and useful mathematicaltools which describe uncertainty but each of them has its own limitation pointedout by Molodstov. Therefore, Molodstov introduced the theory of soft sets tomodel vague and uncertain information [36]. A soft set a parameterized collectionof subsets of a universe of discourse. This mathematical tool is free from parame-terization inadequacy, syndrome of fuzzy set theory, rough set theory, probabilitytheory and so on. Soft set theory has been applied successfully in several areassuch as, smoothness of functions, game theory, operation research, Riemann in-tegration, Perron integration, and probability. Maji et al. gave the application ofsoft sets in decision making problem [32; 33; 34]. Recently soft set theory attainedmuch attention of the researchers since its appearance and start studying soft al-gebraic structures. Aktas and Cagman introduced soft groups which laid downthe foundations to study algebraic structures in the context of soft sets [1]. Someproperties and algebra may be found in [8] : Feng et al. studied soft semigroupsin [24]. Alcantud discussed some formal relationships among soft sets,fuzzy sets, and their extensions [5]. Alcantud et al. and Muthukumarpresented the application of soft sets in medical diagnosis [3; 37]. A hugeamount of literature can be seen in [6; 7; 8; 9; 10; 16; 37; 45; 48] :The main purpose of this paper is to introduce algebraic soft coding theory

which extends the notion of a code to soft sets. A soft code is a parameterizedcollection of codes. Di¤erent types of error correcting codes have been extend toconstruct soft error correcting codes. A variety of soft codes can be found byapplying soft sets to codes. Soft linear codes have been discussed mainly in thispaper. The novel concept of soft dimension have been introduced which in fact ageneralization of the dimension of a code and the concept of soft minimum distanceis introduced here. Soft codes of type 1 have been established in this paper. Theimportant notions of soft generator matrix as well as soft parity check matrix havebeen constructed to study more features of soft linear codes. Further, the notionsof soft complete codes are introduced in this paper and in the end two soft decodingalgorithm has been costructed in this paper.

NEW CLASS OF SOFT LINEAR ALGEBRAIC CODES AND THEIR PROPERTIES USING SOFT SETS3

The organization of this paper is as follows: In section 2, basic concepts of soft sets and codes are presented. In section 3, the important notions of soft codes are given with the study of some of their basic properties and features. In sec- tion 4, soft generator matrix (generator n-matrix) and soft parity check matrix (parity check n-matrix) are presented. Section 5 is about the soft decoding of soft codes. We developed two techniques for the decoding of soft codes. Further, in this section, two examples are presented for the veri�cation of soft decoding process. In section 6, we gave the idea of soft communication process and we also constructed a model for this soft communication process. In section 7, we presented the main disticntion and comparison of soft linear codes with linear codes. Conclusion is given in section 8.

2. Basic concepts

This section has two subsections. First subsection recalls all the basic notionsof linear algebraic codes. The second subsection gives the basic de�nition andproperties of soft sets.

2.1. Codes.

De�nition 1. [30]Let H be an n� k � n matrix with elements in Kq. The set ofall n-dimensional vectors satisfying HxT = (0) over Kq is called a linear code(blockcode) C over Kq of block length n. C is also known as linear (n; k) code.

De�nition 2. [43]. Each vector of the subspace C is termed as codewordof the lenght n. A codeword is used for secrecy or convenience insteadof the usual name for something.

De�nition 3. [43]. Let Kn be a vector space over the �eld K, and x; y 2 Kn wherex = x1x2:::xn,y = y1y2:::yn. The Hamming distance between the vectors x and y isdenoted by d (x; y), and is de�ned as d (x; y) = ji : xi 6= yij.De�nition 4. [43]. The minimum distance of a code C is the smallest distancebetween any two distinct codewords in C which is denoted by d (C) ; that is d (C) =min fd (x; y) : x; y 2 C, x 6= yg :De�nition 5. [43]. Let K be a �nite �eld and n be a positive integer. Let C be asubspace of the vector space V = Kn. Then C is called a linear code over K.

De�nition 6. [43]. The linear code C is called linear [n; k]-code if dim(C) = k.

De�nition 7. [43]. Let C be a linear [n; k]-code. Let G be a k � n matrix whoserows form basis of C: Then G is called generator matrix of the code C.

De�nition 8. [43]. Let C be an [n; k]-code over K. Then the dual code of C isde�ned to be

C? = fy 2 Kn : x � y = 0 for all x 2 CgDe�nition 9. [43]. Let C be an [n; k]-code and let H be the generator matrix ofthe dual code C?. Then H is called a parity-check matrix of the code C:

De�nition 10. [43]. A code C is called self-orthogonal code if C � C?.De�nition 11. [43]. Let C be a code over the �eld K and for every x 2 Kn; thecoset of C is de�ned to be

Cc = fx+ c : c 2 Cg

4 MUMTAZ ALI, FLORENTIN SMARANDACHE, AND W.B. VASANTHA KANDASAMY

De�nition 12. [43]. Let C be a linear code over K: The coset leader of a givencoset C is de�ned to be the vector with least weight in that coset.

De�nition 13. [43]. If a codeword x is transmitted and the vector y is received,then e = y � x is called error vector. Therefore a coset leader is the error vectorfor each vector y lying in that coset.

2.2. Soft set. Throughout this subsection U refers to an initial universe, N is aset of parameters, P (U) is the power set of U , and A � N . Molodtsov [31]. de�nedthe soft set in the following manner:

De�nition 14. [31]. A pair (F;A) is called a soft set over U where F is a mappinggiven by F : A �! P (U).

In other words, a soft set over U is a parameterized family of subsets of theuniverse U . For a 2 A, F (a) may be considered as the set of a-elements of the softset (F;A), or as the set of a-approximate elements of the soft set.

De�nition 15. [28]. For two soft sets (F;A) and (H;B) over U , (F;A) is calleda soft subset of (H;B) if

(1) A � B and(2) F (e) � G(e), for all e 2 A.

This relationship is denoted by (F;A)�� (H;B). Similarly (F;A) is called a

soft superset of (H;B) if (H;B) is a soft subset of (F;A) which is denoted by

(F;A)�� (H;B).

De�nition 16. [28]. Two soft sets (F;A) and (H;B) over U are called soft equalif (F;A) is a soft subset of (H;B) and (H;B) is a soft subset of (F;A).

3. Soft Linear Code

In this section for the �rst time the notion of soft code and soft code of type 1are introduced. Examples of these codes are provided.

De�nition 17. Let K be a �nite �eld and V = Kn be a vector space over K wheren is a positive integer. Let P (V ) be the power set of V and (F;A) be a soft set overV: Then (F;A) is called soft linear code over V if and only if each F (a) is a linearcode (subspace) of V for all a 2 A. In the rest of the paper, from a soft code, wemean a soft linear code (F;A) :

Example 1. Let K = K2 and V = K32 is a vector space over K2 and let (F;A)

be a soft set over V = K32 . Then clearly (F;A) is a soft linear code over V = K

32 ,

where

F (a1) = f000; 111g ;F (a2) = f000; 110; 101; 011g :

De�nition 18. Let (F;A) be a soft code over the �eld K: Then the softelement F (a) is called soft codeword of (F;A) for all a 2 A:Here wedenote the soft codeword by Ys. In other words a soft codeword is aparameterized set of codewords of C.

NEW CLASS OF SOFT LINEAR ALGEBRAIC CODES AND THEIR PROPERTIES USING SOFT SETS5

De�nition 19. Let (F;A) be a soft code over V = Kn. Then Ds is called softdimension of (F;A) if

Ds = fdim (F (a)) ; for all a 2 Ag :

The soft dimension Ds of the soft code is simply an n-tuple; where n is thenumber of parameters in the parameter set A:

Example 2. Let (F;A) be a soft code de�ned in above example. Then the softdimension is as follows,

Ds = fdim (F (a1)) = 1;dim (F (a2)) = 2g ;= f1; 2g :

De�nition 20. A soft linear code (F;A) over V of soft dimension Ds is called softlinear [n;Ds]-code.

De�nition 21. Let (F;A) be a soft code over V . Then the soft minimum distanceof (F;A) is denoted by Sd (F;A) and is de�ned to be

Sd (F;A)= fd (F (a)) : for all a 2 Ag ; where d (F (a)) is the minimum distance of the code F (a) :

Example 3. Let (F;A) be a soft code d�ned in Example 1. Then the min-imum distance of the code F (a1) = 3 and F (a2) = 2. Thus the soft minimumdistance of the soft code (F;A) is given as

Sd (F;A) = fd (F (a1)) = 3; d (F (a2)) = 2g ;= f3; 2g :

De�nition 22. A soft code (F;A) in V over the �eld K is called soft code of type1, if the dimension of F (a) is same, for all a 2 A.

Example 4. Let (F;A) be a soft code in V over the �eld K22 , where

F (a1) = f00; 01g ; F (a2) = f00; 10g :

The soft dimension Ds of (F;A) is as follows,

Ds = f1; 1g :Thus clearly (F;A) is a type 1 soft code.

Theorem 1. Every soft code of type 1 is trivially a soft code but the converse isnot true.

For converse, we take the following example.

Example 5. Let (F;A) be a soft code d�ned in Example 1. Then clearly(F;A) is not a type 1 soft code.

4. Soft Generator Matrix and Soft Parity Check Matrix

In this section the soft generator n-matrix and the parity check n-matrix of thesoft linear code is introduced and examples are given. When n = 2; it reduces tothe bimatrix. In this paper, we used the convnetion of replacing [ by j :

De�nition 23. Let (F;A) be a soft linear [n;Ds]-code. Let Gs be the n-matrixwhose elements are the generator matrices of the soft code (F;A), corresponding toeach a 2 A where n is the number of parameters in A. Then Gs is termed as thesoft generator matrix of the soft linear code (F;A).

6 MUMTAZ ALI, FLORENTIN SMARANDACHE, AND W.B. VASANTHA KANDASAMY

Example 6. Let (F;A) be a soft code de�ned in Example 1. Clearly wehave,

F (a1) has generator matrix GF (a1) =�1 1 1

�and F (a2) has generator

matrix GF (a2) =�1 1 00 1 1

�.

Then the soft generator matrix of the soft code (F;A) is a bimatrix given asfollows.

Gs =

��1 1 1

� ����� 1 1 00 1 1

��:

De�nition 24. Let (F;A) be a soft [n;Ds]-code over the �eld K and the vectorspace V: Then the soft dual code of (F;A) is de�ned to be

(F;A)dual = fF (a)dual : for all a 2 Ag ;

wher F (a)dual is the dual code of F (a).

Example 7. Let (F;A) be a soft code de�ned in Example 1. Then the softdual code of (F;A) is (F;A)dual, where

F (a1)dual = f000; 110; 101; 011g ;F (a2)dual = f000; 111g :

De�nition 25. A soft linear code (F;A) in V = Kn over the �eld K is calledcomplete-soft code if for all a 2 A, the dual of F (a) also exist in (F;A).

Example 8. Let (F;A) be a soft code de�ned in Example 1. Then clearly(F;A) is a complete-soft code because the dual of F (a1) is F (a2) and also the dualof F (a2) is F (a1).

Theorem 2. All complete-soft codes are trivially soft codes but the converse is nottrue in general.

Theorem 3. A complete-soft code (F;A) over the �eld K and the vector space Vis the parametrized collection of the codes C with its dual code C?:

De�nition 26. Let (F;A) be a soft code over the �eld K and the vector space Vand (F;A)dual be the soft dual code of (F;A) :Then the soft dimension of the softdual code (F;A)dual is denoted by (Ds)dual and is de�ned as

(Ds)dual = fdim (F (a)dual) : for all a 2 Ag ; where dim (F (a)dual) is the dimenesion of the dual code F (a) :

Example 9. In previous example the soft dimension of the dual code (F;A)dualis following

(Ds)dual = fdim (F (a1)dual) = 2;dim (F (a1)dual) = 1g= f2; 1g :

De�nition 27. Let (F;A) be a soft over the �eld K and the vector space V and letHs be the soft generator matrix of the soft dual code (F;A)dual : Then Hs is calledthe soft parity check matrix of the soft code (F;A) :

Example 10. In above example the soft dual code of (F;A) is (F;A)dual, where

F (a1)dual = f000; 110; 101; 011g ;F (a2)dual = f000; 111g :

NEW CLASS OF SOFT LINEAR ALGEBRAIC CODES AND THEIR PROPERTIES USING SOFT SETS7

The soft generator bimatrix of (F;A)dual is S (H), where

Hs =

��1 1 00 1 1

� ��� 1 1 1��;

where HF (a1) =�1 1 00 1 1

�is a parity check matrix of GF (a1) and HF (a2) =�

1 1 1�is a parity check matrix of GF (a2).

Theorem 4. Let (F;A) be a soft code over the �eld K and the vector space V . LetGs and Hs be the soft generator matrix and soft parity check matrix of the softcode (F;A) : Then

GsHTs = 0:

De�nition 28. A soft linear code (F;A) in V over the �eld K is called soft selfdual code if (F;A)dual = (F;A) :

De�nition 29. Let (F;A) be a soft code over the �eld K and the vector space V .Let Gs be the soft generator matrix of (F;A) : Then the soft canonical generatormatrix of (F;A) is denoted by G�s and is de�ned as

G�S =hG�F (a)

i;

where G�F (a) is the canonical generator matrix of F (a), for all a 2 A: In fact asoft canonical generator matrix is a n-canonical generator matrix.

Example 11. Let (F;A) be a soft code over V = K52 , where

F (a1) = f00000; 10010; 01001; 00110; 11011; 10100; 01111; 11101g ; F (a2) = f00000; 11111; 10110; 01001g :

The soft canonical generator bimatrix of (F;A) is as under,

G�s =

24� 1 0 1 1 00 1 0 0 1

� ������24 1 0 0 1 00 1 0 0 10 0 1 1 0

3535 ;where G�F (a1) =

�1 0 1 1 00 1 0 0 1

�; G�F (a2) =

24 1 0 0 1 00 1 0 0 10 0 1 1 0

35 are respec-tively the canonical generator matrices of F (a1) and F (a2) :

Theorem 5. Let (F;A) be a soft code. If (F;A) has a soft canonical generatorn-matrix

G�s =hG�F (a) =

hIk... A

i, for all a 2 A

i:

Then

H�s =

�H�F (a) =

��AT

... Im�k

�, for all a 2 A

�is the soft canonical parity check n-matrix of (F;A). Conversely, if

H�s =

hH�F (a) =

hB... Im�k

i, for all a 2 A

iis the soft canonical parity check matrix of (F;A), then

G�s =hG�F (a) =

hIk... �BT

i, for all a 2 A

i:

8 MUMTAZ ALI, FLORENTIN SMARANDACHE, AND W.B. VASANTHA KANDASAMY

5. Soft Decoding Algorithms

In this section we provide a soft decoding algorithm for soft linear codes whichforms subsection one of this section. This concept is illustrated with examples. Insubsection two soft syndrome decoding is introduced. This helps in error correctionof soft codes.

5.1. Soft Standard Array Decoding.

De�nition 30. Let (F;A) be a soft linear code or soft subspace of V = Kn overthe �eld K. Then for every x 2 Kn, the soft coset of (F;A) is de�ned asfollowing.

(F;A)C = fx+ F (a) , for all a 2 Ag :

The soft coset of (F;A) is denoted by (F;A)C .

De�nition 31. Let (F;A) be a soft linear code in V = Kn over the �eld K: Thesoft coset leader of a given soft coset (F;A)C is denoted by Ei and is de�ned to be

Ei = fu : for all a 2 Ag ;where u is the coset leader of F (a) :

Example 12. Let (F;A) be a soft code de�ned in Example 1. Then the softcosets of the soft code (F;A) are as follows:

(F;A)C1 = f000 + F (a1) ; 000 + F (a2)g = (F;A) ;

(F;A)C2 = f100 + F (a1) ; 100 + F (a2)g = ff100; 011g ; f100; 010; 001; 111gg ;

(F;A)C3 = f010 + F (a1) ; 010 + F (a2)g = ff010; 101g ; f100; 010; 001; 111gg ;

(F;A)C4 = f001 + F (a1) ; 001 + F (a2)g = ff001; 110g ; f100; 010; 001; 111gg :Following are the soft coset leaders of the soft coset (F;A),

E1= f000; 000g ; E

2= f100; 100g ;

E3= f010; 010g ; E

4= f001; 001g ; and so on.

De�nition 32. A set of standard arrays of the corresponding parametrized codesis called soft standard array for the soft linear code (F;A).

Example 13. Let (F;A) be a soft code de�ned in Example 1. Then the softcosets of (F;A) are as follows.

(F;A)C1 = f000 + F (a1) ; 000 + F (a2)g = (F;A) ;

(F;A)C2 = f100 + F (a1) ; 100 + F (a2)g = ff100; 011g ; f100; 010; 001; 111gg ;

(F;A)C3 = f010 + F (a1) ; 010 + F (a2)g = ff010; 101g ; f100; 010; 001; 111gg ;

NEW CLASS OF SOFT LINEAR ALGEBRAIC CODES AND THEIR PROPERTIES USING SOFT SETS9

(F;A)C4 = f001 + F (a1) ; 001 + F (a2)g = ff001; 110g ; f100; 010; 001; 111gg :The following are the coset leaders of the soft linear code (F;A),

E1= f000; 000g ; E

2= f100; 100g ;

E3= f010; 010g ; E

4= f001; 001g :

The soft standard array for the soft linear code (F;A) is following,8>><>>:000 111100 011010 101001 110

;

000 110 101 011100 010 001 111010 100 111 001001 111 100 010

9>>=>>; :5.2. Soft Syndrome Decoding.

De�nition 33. Let (F;A) be a soft code over K with soft parity check matrix HS.For any vector Ys � Kn, the soft syndrome of Ys is de�ned as S (Ys) = Ys (Hs)

T .

Theorem 6. Let (F;A) be a soft code over the �eld K: For Ys � Kn, the softcodeword nearest to Ys is given by Xs = Ys �Es, where Es is the soft coset leader.

Let (F;A) be a soft code over the �eld K with soft parity check matrix HS : Forsoft syndrome decoding, we �rst �nd all the soft coset of the soft code (F;A) andthen �nd the soft coset leaders ES which are in fact the collection of coset leaderscorresponding to each parameterized code. Then, we compute the soft syndromefor all the soft coset leaders and then make a table of soft coset leaders with theirsoft syndroms. To decode a soft codeword say YS , we simply �nd the soft syndromeof that soft codeword and then compare their soft syndrome with soft coset leadersyndrome. After comparing their soft syndromes, we then subtract the soft cosetleader from the soft decoded word. Hence YS is soft decoded as XS = YS � Es.

Example 14. Let (F;A) be a soft code de�ned in Example 1. The soft paritycheck matrix of (F;A) is Hs, where

Hs =

��1 1 00 1 1

� ��� 1 1 1��:

The transpose of Hs is following,

(Hs)T=

2424 1 01 10 1

35 ������24 111

3535 :Then the soft cosets of (F;A) are as follows.

(F;A)C1 = f000 + F (a1) ; 000 + F (a2)g = (F;A) ;

(F;A)C2 = f100 + F (a1) ; 100 + F (a2)g = ff100; 011g ; f100; 010; 001; 111gg ;

(F;A)C3 = f010 + F (a1) ; 010 + F (a2)g = ff010; 101g ; f100; 010; 001; 111gg ;

10 MUMTAZ ALI, FLORENTIN SMARANDACHE, AND W.B. VASANTHA KANDASAMY

(F;A)C4 = f001 + F (a1) ; 001 + F (a2)g = ff001; 110g ; f100; 010; 001; 111gg :

The following are the soft coset leaders Ei of the soft code (F;A),

E1= f000; 000g ; E

2= f100; 100g ;

E3 = f010; 010g ; E4 = f001; 001g :Soft syndrome table of soft coset leader is given as follows.

soft coset leader soft syndromef000; 000g f00; 0gf100; 100g f10; 1gf010; 010g f11; 1gf001; 001g f01; 1g

We want to decode a soft codeword Ys = f110; 101g. First we �nd the softsyndrom of the soft codeword.

S (Ys) = y (Hs)T= [110; 100]

2424 1 01 10 1

35 ������24 111

3535 = [01; 1] :Since S (f110; 100g) = f01; 1g = S (f001; 001g) :The decoded soft codeword is

S (Ys)� S (E4) = Xs

S (f110; 100g)� S (f001; 001g) = f110� 001; 100� 001g= f111; 101g :

Hence the soft codeword Ys = f110; 100g is decoded as Xs = f111; 101g : Bysimilar fashion, we can �nd all the soft decoded codewords.

6. Soft Communication Transmission

In this section, we present a soft communication transmission with the help ofa model. This model consists of a soft encoder which is the collection of encoders.Therefore, corresponding to each parameter a in A, we have an encoder in thesoft encoder. Furthermore, we have also a soft decoder which is the collection ofdecoders and thus to each parameter a in A, we have a decoder in the soft decoder.There are n parameters in the parameter set A, therefore, we have n encoders inthe soft encoder as well as n decoders in the soft decoder. Consequently, we havea collection of n messages and n receivers to recieve their desired message.The signals from the source cannot be transmitted directly by the channel.

Therefore the n encoders perform the important work of data reduction and suit-ably transforms the n-messages into usable form. Thus there is a di¤erence betweensource encoding and channel encoding. The former reduces the messages to recog-nizable parts and the latter adds redundant information to enable deduction andcorrection of possible errors in transmission. Similarly on receiving we have todistinguish between channel decoding and source decoding, which invert the corre-sponding channel and source encoding besides deleting and correcting errors. If we

NEW CLASS OF SOFT LINEAR ALGEBRAIC CODES AND THEIR PROPERTIES USING SOFT SETS11

have n = 1, the soft communication transmission reduces to classical communica- tion transmission. The model of soft communication transmission is given in the following Figure 1.

Figure1 : Soft communication transmission model

7. Comparison of Soft Linear Codes with linear Codes

In this section, we have presented some of the main di¤erences of soft linearcodes with linear codes by comparison. The following are the main distinguishingfeatures of our proposed soft linear codes.

(1) The main distinction between linear code and soft linear code is that forthe soft linear code each soft codeword has some �avors, i.e. each softcodeword is characterized by some attributes, while the non-soft codewordsare characterized by no attributes. Then one can manipulate the attributesof a soft codewords, for example an attribute "a1" can some some attributethat �trick�the code hackers, or may be chances that the soft codeword hasa smaller chance to belong to the message (i.e. included just to deceivethe hackers). and so on. Hence soft linear codes are more secure than theclassical codes due to parameterization.

(2) The soft linear code di¤er in structure as a linear code C is just a onesubspace, but a soft linear code is a collection of subspaces. At least a

12 MUMTAZ ALI, FLORENTIN SMARANDACHE, AND W.B. VASANTHA KANDASAMY

soft linear code has two subspaces, each subspace depending on the set ofparameters used. Therefore, soft linear codes are more generalized than thelinear codes.

(3) Soft linear codes can simultaneously transmit n distinct messages to n set ofreceivers whereas linear codes can transmit only one message to a receiver.

(4) This new technique makes use of bi-matrices or to be more general usesthe concept of n-matrices. Certainly this notion will save both time andeconomy.

(5) In soft decoding process, one can decode a set of code words (soft codeword) at a time while it is not possible in linear decoding process.

(6) Soft linear codes can provide us several types of new codes such as type-1codes, complete codes etc., which is not possible in linear codes.

8. Conclusion

Algebraic codes play a signi�cant role in the minimization of data corruptionwhich caused by defects such as inference, noise channel, crosstalk, and packetlost. In this paper for the �rst time we have introduced the new notions of softlinear codes using soft sets. The advantages of this new class of codes are that itcan send simultaneously n-messages to n-persons. Hence this new codes can saveboth time and economy. Soft generator matrix (generator n-matrix) and soft paritycheck matrix (parity check n-matrix) are presented. Two techniques are developedfor the decoding of soft linear codes. The channel transmission is also illustrated.Finally, the main disticntion and comparison of soft linear codes with linear codesare presented.

Acknowledgement 1. There is no con�ict of interest of the authors and the workdescribed in this paper has neither published nor under consideration for publishing.

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Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320,PakistanE-mail address : mumtazali7288@gmail.com

University of New Mexico, 705 Gurley Ave., Gallup, New Mexico 87301, USAE-mail address : fsmarandache@gmail.com

Department of Mathematics Indian Institute of Technology Madras, Chennai �600036,India

E-mail address : vasanthakandasamy@gmail.com